Indices

Index laws are used throughout this chapter when simplifying and solving exponential expressions. A solid grasp of these rules will make working with exponentials and logarithms much more straightforward.

Technique: Laws of indices

For any base $a>0$ and indices $m,$ $n$:

• $a^m\times a^n=a^{m+n}$
• $a^m÷a^n=a^{m-n}$
• $(a^m{)}^n=a^{mn}$
• $a^n\times b^n=(ab{)}^n$

For example

• $2^3\cdot 2^5=2^8$
• ${\displaystyle \frac{2^3}{2^5}=2^{-2}}$
• ${\left(2^3\right)}^5=2^{15}$

Technique: Special indices

• $a^0=1$
• $a^{-n}={\displaystyle \frac{1}{a^n}}$
• $a^{\frac{1}{2}}=\sqrt{a}$
• $a^{1/n}=\sqrt[n]{a}$
• $a^{m/n}={\left(\sqrt[n]{a}\right)}^m$

For example,

• $3^0=1$
• $3^{-1}={\displaystyle \frac{1}{3}}$
• $3^{-2}={\displaystyle \frac{1}{3^2}}={\displaystyle \frac{1}{9}}$
• $3^{1/2}=\sqrt{3}$